A lower bound on the analytic log-canonical threshold over local fields of positive characteristic

Given a local field $F$ of positive characteristic, an $F$-analytic manifold $X$ and an analytic function $f:X\to F$, the $F$-analytic log-canonical threshold ${\mathrm{l}\mathrm{c}\mathrm{t}}_F(f;x_0)$ is the supremum over the values $s\ge 0$ such that ${\mid f\mid }_F^{-s}$ is integrable near $x_0\in X$. We show that ${\mathrm{l}\mathrm{c}\mathrm{t}}_F(f;x_0)>0$. Moreover, if $f$ is a regular function on a smooth algebraic $F$-variety, we obtain an effective lower bound ${\mathrm{l}\mathrm{c}\mathrm{t}}_F(f;x_0)>C$, where $C>0$ is explicit and depends only on the complexity class of $X$ and $f$.